5 The following diagram shows a simplified model for an auto

5. The following diagram\' shows a simplified model for an automobile suspension-the \"quarter car model.\" mass, M Figure 632 Dagrammatic representatin ot an automotive suspension syilern. Here, represents the distance between the chassis and the road turtace when the automobile st rest. rn y the potition the chassis above the reference elevation, and ain the arvatin of he road above the reference tevation. Oppenheim and Willsky, page 474.

Solution

solution:

1) here input from support is x(t)=Xsinwt

x\'(t)=Xwcoswt

x\'\'(t)=-Xw2sinwt

and it will produce periodic output

y(t)=Ysinwt

where

y\'(t)=Ywcoswt

y\'\'(t)=-Yw2sinwt

2) here absolute deflection of spring is =(y-x)

absolute velocity=(y\'-x\')

where FBD of mass has spring force,damping force ,inertia acting vertically upward,hence equation of motion is

my\'\'+cy\'+ky=cx\'+kx

putting value in equation we get

my\'\'+cy\'+ky=X(ksinwt+ccoswt)

as term in bracket is in triangle law hence we can write

my\'\'+cy\'+ky=X(k2+c2w2)^.5*(cosasinwt+sinacoswt)

where

cosa=k/(k2+c2w2)^.5 and sina=cw/(k2+c2w2)^.5

hence equation is

my\'\'+cy\'+ky=X(k2+c2w2)^.5*(sinwt+a)

this equation is similar to

mx\'\'+cx\'+kx\'=F0sinwt

where F0=X(k2+c2w2)^.5

where solution is given by

y=yc+yp

yc=Y(exp^-(zeta*wn*t))sin(wdt+phi)

where yp=Ysin(wt+a-phi)

3) hence on solving steady state amplitude of vibration iss given by

Y=X*((k2+c2w2)^.5/k)/(((1-(w/wn)2)2+(2*zeta*(w/wn))2)^.5)

4) where transfer function which is ratio of output y(t) to input x(t) is given as

H(t)=y(t)/x(t)

H(t)=Ysinwt/Xsinwt

H(t)=Y/X=((k2+c2w2)^.5/k)/(((1-(w/wn)2)2+(2*zeta*(w/wn))2)^.5)

5)to simulate system let consider w=wn,zeta=0 low value,and input amplitude ofX= .1 m

so we get

Y=infinite

hence for low value of damping vehicla produce to unpleasant ride and unpleasant ride turn pleasant with addition of damper in the sytsem

6) system produce pleasant ride if damping factor is within underdamped condition and value of spring stiffness k should be high enough, mass M should be low and road condition has to be good means w of road input should be high enough means its amplitude should be as small as possible

hence zeta<1,

K as high as possible,

M=low

w=low frequency of disturbance

good ride means when relative amplitude od mass is zero,that is

z(t)=y(t)-x(t)=0